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<a href="_trust_region_newton_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno">    1</span><span class="comment">/*!</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno">    2</span><span class="comment"> * </span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno">    3</span><span class="comment"> *</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno">    4</span><span class="comment"> * \brief       Trust-Region Newton-Step Method</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno">    5</span><span class="comment"> *</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno">    6</span><span class="comment"> * \author      O. Krause</span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno">    7</span><span class="comment"> * \date        2015</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno">    8</span><span class="comment"> *</span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno">    9</span><span class="comment"> *</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno">   10</span><span class="comment"> * \par Copyright 1995-2017 Shark Development Team</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno">   11</span><span class="comment"> * </span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno">   12</span><span class="comment"> * &lt;BR&gt;&lt;HR&gt;</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno">   13</span><span class="comment"> * This file is part of Shark.</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno">   14</span><span class="comment"> * &lt;https://shark-ml.github.io/Shark/&gt;</span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno">   15</span><span class="comment"> * </span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno">   16</span><span class="comment"> * Shark is free software: you can redistribute it and/or modify</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno">   17</span><span class="comment"> * it under the terms of the GNU Lesser General Public License as published </span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno">   18</span><span class="comment"> * by the Free Software Foundation, either version 3 of the License, or</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno">   19</span><span class="comment"> * (at your option) any later version.</span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno">   20</span><span class="comment"> * </span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno">   21</span><span class="comment"> * Shark is distributed in the hope that it will be useful,</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno">   22</span><span class="comment"> * but WITHOUT ANY WARRANTY; without even the implied warranty of</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno">   23</span><span class="comment"> * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno">   24</span><span class="comment"> * GNU Lesser General Public License for more details.</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno">   25</span><span class="comment"> * </span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno">   26</span><span class="comment"> * You should have received a copy of the GNU Lesser General Public License</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno">   27</span><span class="comment"> * along with Shark.  If not, see &lt;http://www.gnu.org/licenses/&gt;.</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno">   28</span><span class="comment"> *</span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno">   29</span><span class="comment"> */</span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno">   30</span> </div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno">   31</span><span class="preprocessor">#ifndef ALGORITHMS_GRADIENTDESCENT_TRUST_REGION_NEWTON_H</span></div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno">   32</span><span class="preprocessor">#define ALGORITHMS_GRADIENTDESCENT_TRUST_REGION_NEWTON_H</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno">   33</span> </div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno">   34</span><span class="preprocessor">#include &lt;<a class="code" href="_d_l_l_support_8h.html">shark/Core/DLLSupport.h</a>&gt;</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno">   35</span><span class="preprocessor">#include &lt;<a class="code" href="_abstract_single_objective_optimizer_8h.html">shark/Algorithms/AbstractSingleObjectiveOptimizer.h</a>&gt;</span></div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno">   36</span> </div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno">   37</span><span class="keyword">namespace </span><a class="code hl_namespace" href="namespaceshark.html" title="AbstractMultiObjectiveOptimizer.">shark</a> {</div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno">   38</span><span class="comment"></span> </div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno">   39</span><span class="comment">/// \brief Simple Trust-Region method based on the full Hessian matrix</span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno">   40</span><span class="comment">///</span></div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno">   41</span><span class="comment">/// While normal Newton methods compute the Newton steps and perform a line-search</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno">   42</span><span class="comment">/// In the Newton direction, trust region methods first choose a maximal step-length and</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno">   43</span><span class="comment">/// then try to find an approximate best point of the second order tailor expansion in that</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno">   44</span><span class="comment">/// region. more formally, we solve</span></div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno">   45</span><span class="comment">/// \f[ \min_{p} m(p) = p^T B p +g^Tp, ||p||&lt;\delta \f]</span></div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno">   46</span><span class="comment">/// where \f$B\f$ is the Hessian and \f$g\f$ the gradient of the current point \f$x\f$.</span></div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno">   47</span><span class="comment">/// Given this step, we compute how much the model agrees with the actual function, i.e.</span></div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno">   48</span><span class="comment">/// \f[ \rho = \frac{ f(x+p)-f(p) }{m(p)} \f]</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno">   49</span><span class="comment">/// If this value is large, that is, the improvement in function value is approximately as</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno">   50</span><span class="comment">/// large or larger as the model predicted, we increase \f$\delta\f$ to make larger steps</span></div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno">   51</span><span class="comment">/// possible, otherwise we decrease it, if the model predicted a much larger improvement</span></div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno">   52</span><span class="comment">/// than observed - in the worst case, the new point is worse than the old one.</span></div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno">   53</span><span class="comment">/// </span></div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno">   54</span><span class="comment">/// As a further check, to improve convergence, we do not accept every step, but those</span></div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno">   55</span><span class="comment">/// with \f$ \rho &gt; c &gt; 0 \f$. This ensures that we do not overjump the optimum too much</span></div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno">   56</span><span class="comment">/// and leads to a better (worst case) convergence rate.</span></div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno">   57</span><span class="comment">///</span></div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno">   58</span><span class="comment">/// The optimal step is computed by a conjugate gradient method that stops once a</span></div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno">   59</span><span class="comment">/// target tolerance is reached, or the step approaches the boundary (which happens,</span></div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno">   60</span><span class="comment">/// for example, when the Hessian is indefinite or rank-deficient). Thus, computation</span></div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno">   61</span><span class="comment">/// time is not wasted for steps that are far away from the optimum. The tolerance</span></div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno">   62</span><span class="comment">/// is set by a forcing-schedule so that accuracy increases in the vicinity of the</span></div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno">   63</span><span class="comment">/// optimum, enabling solutions with arbitrary precision.</span></div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno">   64</span><span class="comment">///</span></div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno">   65</span><span class="comment">/// The algorithm is based on </span></div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno">   66</span><span class="comment">/// Jorge Nocedal, Stephen J. Wright</span></div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno">   67</span><span class="comment">/// Numerical Optimization, 2nd Edition</span></div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno">   68</span><span class="comment">/// Algorithm 4.1 with Algorithm 7.2 to solve the sub-problem</span></div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno">   69</span><span class="comment">/// \ingroup gradientopt</span></div>
<div class="foldopen" id="foldopen00070" data-start="{" data-end="};">
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html">   70</a></span><span class="comment"></span><span class="keyword">class </span><a class="code hl_class" href="classshark_1_1_trust_region_newton.html" title="Simple Trust-Region method based on the full Hessian matrix.">TrustRegionNewton</a> : <span class="keyword">public</span> <a class="code hl_class" href="classshark_1_1_abstract_single_objective_optimizer.html" title="Base class for all single objective optimizer.">AbstractSingleObjectiveOptimizer</a>&lt;RealVector &gt;</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno">   71</span>{</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno">   72</span><span class="keyword">public</span>:<span class="comment"></span></div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno">   73</span><span class="comment">    /// \brief Default constructor.</span></div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#ab4bf5d1daf70ee438d8cbdfa258c3029">   74</a></span><span class="comment"></span>    <a class="code hl_define" href="_d_l_l_support_8h.html#a54b73283f7f70b27fbd8ac5d4621827f" title="Defines SHARK_COMPILE_DLL.">SHARK_EXPORT_SYMBOL</a> <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#ab4bf5d1daf70ee438d8cbdfa258c3029" title="Default constructor.">TrustRegionNewton</a>();</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno">   75</span><span class="comment"></span> </div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno">   76</span><span class="comment">    /// \brief Initialize the iterative optimizer with a problem (objective function) and a starting point.</span></div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno">   77</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno">   78</span><span class="comment">    /// The initial trust region radius is set to 0.1</span></div>
<div class="foldopen" id="foldopen00079" data-start="{" data-end="}">
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#a891852f57ed31dda460a76deae33d516">   79</a></span><span class="comment"></span>    <span class="keywordtype">void</span> <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#a891852f57ed31dda460a76deae33d516" title="Initialize the iterative optimizer with a problem (objective function) and a starting point.">init</a>(<a class="code hl_typedef" href="classshark_1_1_abstract_single_objective_optimizer.html#aa4c05609c54d7ebc99d099e7dd6e228f">ObjectiveFunctionType</a>&amp; objectiveFunction, <a class="code hl_typedef" href="classshark_1_1_abstract_single_objective_optimizer.html#a85f0d04fdfb094dba4dc80b1fb5e3adb">SearchPointType</a> <span class="keyword">const</span>&amp; startingPoint){</div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno">   80</span>        <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#a891852f57ed31dda460a76deae33d516" title="Initialize the iterative optimizer with a problem (objective function) and a starting point.">init</a>(objectiveFunction,startingPoint,0.1);</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno">   81</span>    }<span class="comment"></span></div>
</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno">   82</span><span class="comment">    /// \brief Initialize the iterative optimizer with a problem (objective function), a starting point and an initial value for the trust-region</span></div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#a3cc77261a8cc3367b87bd60abb1812c7">   83</a></span><span class="comment"></span>    <a class="code hl_define" href="_d_l_l_support_8h.html#a54b73283f7f70b27fbd8ac5d4621827f" title="Defines SHARK_COMPILE_DLL.">SHARK_EXPORT_SYMBOL</a> <span class="keywordtype">void</span> <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#a3cc77261a8cc3367b87bd60abb1812c7" title="Initialize the iterative optimizer with a problem (objective function), a starting point and an initi...">init</a>(<a class="code hl_typedef" href="classshark_1_1_abstract_single_objective_optimizer.html#aa4c05609c54d7ebc99d099e7dd6e228f">ObjectiveFunctionType</a> <span class="keyword">const</span>&amp; objectiveFunction, <a class="code hl_typedef" href="classshark_1_1_abstract_single_objective_optimizer.html#a85f0d04fdfb094dba4dc80b1fb5e3adb">SearchPointType</a> <span class="keyword">const</span>&amp; startingPoint,<span class="keywordtype">double</span> initialDelta);</div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno">   84</span>    <span class="keyword">using </span><a class="code hl_class" href="classshark_1_1_abstract_single_objective_optimizer.html" title="Base class for all single objective optimizer.">AbstractSingleObjectiveOptimizer</a>&lt;RealVector &gt;<a class="code hl_function" href="classshark_1_1_trust_region_newton.html#a891852f57ed31dda460a76deae33d516" title="Initialize the iterative optimizer with a problem (objective function) and a starting point.">::init</a>;</div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno">   85</span>    <span class="comment"></span></div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno">   86</span><span class="comment">    /// \brief From INameable: return the class name.</span></div>
<div class="foldopen" id="foldopen00087" data-start="{" data-end="}">
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#a3b316c4bfc827853a64b03d34c07c165">   87</a></span><span class="comment"></span>    std::string <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#a3b316c4bfc827853a64b03d34c07c165" title="From INameable: return the class name.">name</a>()<span class="keyword"> const</span></div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno">   88</span><span class="keyword">    </span>{ <span class="keywordflow">return</span> <span class="stringliteral">&quot;TrustRegionNewton&quot;</span>; }</div>
</div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno">   89</span><span class="comment"></span> </div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno">   90</span><span class="comment">    /// \brief Minimal improvement ratio (see the algorithm details in the class description).</span></div>
<div class="foldopen" id="foldopen00091" data-start="{" data-end="}">
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#ab8f33456dca90c87be30f3dea462c123">   91</a></span><span class="comment"></span>    <span class="keywordtype">double</span> <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#ab8f33456dca90c87be30f3dea462c123" title="Minimal improvement ratio (see the algorithm details in the class description).">minImprovementRatio</a>()<span class="keyword">const</span>{</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno">   92</span>        <span class="keywordflow">return</span> <a class="code hl_variable" href="classshark_1_1_trust_region_newton.html#a72c8b9f7c03fa28b9d06d2d35aed3280" title="Minimal improvement ratio (see the algorithm details in the class description).">m_minImprovementRatio</a>;</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno">   93</span>    }</div>
</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno">   94</span>    <span class="comment"></span></div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno">   95</span><span class="comment">    /// \brief Minimal improvement ratio (see the algorithm details in the class description).</span></div>
<div class="foldopen" id="foldopen00096" data-start="{" data-end="}">
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#a1bf2c2a5279196f90fe46bcd31c1e8bf">   96</a></span><span class="comment"></span>    <span class="keywordtype">double</span>&amp; <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#a1bf2c2a5279196f90fe46bcd31c1e8bf" title="Minimal improvement ratio (see the algorithm details in the class description).">minImprovementRatio</a>(){</div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno">   97</span>        <span class="keywordflow">return</span> <a class="code hl_variable" href="classshark_1_1_trust_region_newton.html#a72c8b9f7c03fa28b9d06d2d35aed3280" title="Minimal improvement ratio (see the algorithm details in the class description).">m_minImprovementRatio</a>;</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno">   98</span>    }</div>
</div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno">   99</span><span class="comment"></span> </div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno">  100</span><span class="comment">    /// \brief Perform one trust region Newton step, update point and trust region radius.</span></div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#aad95e74aa24a400a20829645a65615f3">  101</a></span><span class="comment"></span>    <a class="code hl_define" href="_d_l_l_support_8h.html#a54b73283f7f70b27fbd8ac5d4621827f" title="Defines SHARK_COMPILE_DLL.">SHARK_EXPORT_SYMBOL</a> <span class="keywordtype">void</span> <a class="code hl_function" href="classshark_1_1_trust_region_newton.html#aad95e74aa24a400a20829645a65615f3" title="Perform one trust region Newton step, update point and trust region radius.">step</a>(<a class="code hl_typedef" href="classshark_1_1_abstract_single_objective_optimizer.html#aa4c05609c54d7ebc99d099e7dd6e228f">ObjectiveFunctionType</a> <span class="keyword">const</span>&amp; objectiveFunction);</div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno">  102</span> </div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno">  103</span><span class="keyword">protected</span>:</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#a303add905011b4cf678b4c3eb204ac1b">  104</a></span>    <span class="keywordtype">double</span> <a class="code hl_variable" href="classshark_1_1_trust_region_newton.html#a303add905011b4cf678b4c3eb204ac1b" title="Current trust region size.">m_delta</a>;                                               <span class="comment">///&lt; Current trust region size</span></div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#a72c8b9f7c03fa28b9d06d2d35aed3280">  105</a></span>    <span class="keywordtype">double</span> <a class="code hl_variable" href="classshark_1_1_trust_region_newton.html#a72c8b9f7c03fa28b9d06d2d35aed3280" title="Minimal improvement ratio (see the algorithm details in the class description).">m_minImprovementRatio</a>;                                 <span class="comment">///&lt; Minimal improvement ratio (see the algorithm details in the class description).</span></div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"><a class="line" href="classshark_1_1_trust_region_newton.html#a57a6384549c0b1bdf613817c3664cbbc">  106</a></span>    ObjectiveFunctionType::SecondOrderDerivative <a class="code hl_variable" href="classshark_1_1_trust_region_newton.html#a57a6384549c0b1bdf613817c3664cbbc" title="First and second derivative of the objective function in the current point.">m_derivatives</a>;   <span class="comment">///&lt; First and second derivative of the objective function in the current point.</span></div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno">  107</span>};</div>
</div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno">  108</span>}</div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno">  109</span><span class="preprocessor">#endif</span></div>
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